3 * Mesa 3-D graphics library
6 * Copyright (C) 1999-2001 Brian Paul All Rights Reserved.
8 * Permission is hereby granted, free of charge, to any person obtaining a
9 * copy of this software and associated documentation files (the "Software"),
10 * to deal in the Software without restriction, including without limitation
11 * the rights to use, copy, modify, merge, publish, distribute, sublicense,
12 * and/or sell copies of the Software, and to permit persons to whom the
13 * Software is furnished to do so, subject to the following conditions:
15 * The above copyright notice and this permission notice shall be included
16 * in all copies or substantial portions of the Software.
18 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
19 * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
21 * BRIAN PAUL BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN
22 * AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
23 * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
28 * eval.c was written by
29 * Bernd Barsuhn (bdbarsuh@cip.informatik.uni-erlangen.de) and
30 * Volker Weiss (vrweiss@cip.informatik.uni-erlangen.de).
32 * My original implementation of evaluators was simplistic and didn't
33 * compute surface normal vectors properly. Bernd and Volker applied
34 * used more sophisticated methods to get better results.
40 #include "main/glheader.h"
41 #include "main/config.h"
44 static GLfloat inv_tab
[MAX_EVAL_ORDER
];
49 * Horner scheme for Bezier curves
51 * Bezier curves can be computed via a Horner scheme.
52 * Horner is numerically less stable than the de Casteljau
53 * algorithm, but it is faster. For curves of degree n
54 * the complexity of Horner is O(n) and de Casteljau is O(n^2).
55 * Since stability is not important for displaying curve
56 * points I decided to use the Horner scheme.
58 * A cubic Bezier curve with control points b0, b1, b2, b3 can be
61 * (([3] [3] ) [3] ) [3]
62 * c(t) = (([0]*s*b0 + [1]*t*b1)*s + [2]*t^2*b2)*s + [3]*t^2*b3
65 * where s=1-t and the binomial coefficients [i]. These can
66 * be computed iteratively using the identity:
69 * [i] = (n-i+1)/i * [i-1] and [0] = 1
74 _math_horner_bezier_curve(const GLfloat
* cp
, GLfloat
* out
, GLfloat t
,
75 GLuint dim
, GLuint order
)
77 GLfloat s
, powert
, bincoeff
;
81 bincoeff
= (GLfloat
) (order
- 1);
84 for (k
= 0; k
< dim
; k
++)
85 out
[k
] = s
* cp
[k
] + bincoeff
* t
* cp
[dim
+ k
];
87 for (i
= 2, cp
+= 2 * dim
, powert
= t
* t
; i
< order
;
88 i
++, powert
*= t
, cp
+= dim
) {
89 bincoeff
*= (GLfloat
) (order
- i
);
90 bincoeff
*= inv_tab
[i
];
92 for (k
= 0; k
< dim
; k
++)
93 out
[k
] = s
* out
[k
] + bincoeff
* powert
* cp
[k
];
96 else { /* order=1 -> constant curve */
98 for (k
= 0; k
< dim
; k
++)
104 * Tensor product Bezier surfaces
106 * Again the Horner scheme is used to compute a point on a
107 * TP Bezier surface. First a control polygon for a curve
108 * on the surface in one parameter direction is computed,
109 * then the point on the curve for the other parameter
110 * direction is evaluated.
112 * To store the curve control polygon additional storage
113 * for max(uorder,vorder) points is needed in the
118 _math_horner_bezier_surf(GLfloat
* cn
, GLfloat
* out
, GLfloat u
, GLfloat v
,
119 GLuint dim
, GLuint uorder
, GLuint vorder
)
121 GLfloat
*cp
= cn
+ uorder
* vorder
* dim
;
122 GLuint i
, uinc
= vorder
* dim
;
124 if (vorder
> uorder
) {
126 GLfloat s
, poweru
, bincoeff
;
129 /* Compute the control polygon for the surface-curve in u-direction */
130 for (j
= 0; j
< vorder
; j
++) {
131 GLfloat
*ucp
= &cn
[j
* dim
];
133 /* Each control point is the point for parameter u on a */
134 /* curve defined by the control polygons in u-direction */
135 bincoeff
= (GLfloat
) (uorder
- 1);
138 for (k
= 0; k
< dim
; k
++)
139 cp
[j
* dim
+ k
] = s
* ucp
[k
] + bincoeff
* u
* ucp
[uinc
+ k
];
141 for (i
= 2, ucp
+= 2 * uinc
, poweru
= u
* u
; i
< uorder
;
142 i
++, poweru
*= u
, ucp
+= uinc
) {
143 bincoeff
*= (GLfloat
) (uorder
- i
);
144 bincoeff
*= inv_tab
[i
];
146 for (k
= 0; k
< dim
; k
++)
148 s
* cp
[j
* dim
+ k
] + bincoeff
* poweru
* ucp
[k
];
152 /* Evaluate curve point in v */
153 _math_horner_bezier_curve(cp
, out
, v
, dim
, vorder
);
155 else /* uorder=1 -> cn defines a curve in v */
156 _math_horner_bezier_curve(cn
, out
, v
, dim
, vorder
);
158 else { /* vorder <= uorder */
163 /* Compute the control polygon for the surface-curve in u-direction */
164 for (i
= 0; i
< uorder
; i
++, cn
+= uinc
) {
165 /* For constant i all cn[i][j] (j=0..vorder) are located */
166 /* on consecutive memory locations, so we can use */
167 /* horner_bezier_curve to compute the control points */
169 _math_horner_bezier_curve(cn
, &cp
[i
* dim
], v
, dim
, vorder
);
172 /* Evaluate curve point in u */
173 _math_horner_bezier_curve(cp
, out
, u
, dim
, uorder
);
175 else /* vorder=1 -> cn defines a curve in u */
176 _math_horner_bezier_curve(cn
, out
, u
, dim
, uorder
);
181 * The direct de Casteljau algorithm is used when a point on the
182 * surface and the tangent directions spanning the tangent plane
183 * should be computed (this is needed to compute normals to the
184 * surface). In this case the de Casteljau algorithm approach is
185 * nicer because a point and the partial derivatives can be computed
186 * at the same time. To get the correct tangent length du and dv
187 * must be multiplied with the (u2-u1)/uorder-1 and (v2-v1)/vorder-1.
188 * Since only the directions are needed, this scaling step is omitted.
190 * De Casteljau needs additional storage for uorder*vorder
191 * values in the control net cn.
195 _math_de_casteljau_surf(GLfloat
* cn
, GLfloat
* out
, GLfloat
* du
,
196 GLfloat
* dv
, GLfloat u
, GLfloat v
, GLuint dim
,
197 GLuint uorder
, GLuint vorder
)
199 GLfloat
*dcn
= cn
+ uorder
* vorder
* dim
;
200 GLfloat us
= 1.0F
- u
, vs
= 1.0F
- v
;
202 GLuint minorder
= uorder
< vorder
? uorder
: vorder
;
203 GLuint uinc
= vorder
* dim
;
204 GLuint dcuinc
= vorder
;
206 /* Each component is evaluated separately to save buffer space */
207 /* This does not drasticaly decrease the performance of the */
208 /* algorithm. If additional storage for (uorder-1)*(vorder-1) */
209 /* points would be available, the components could be accessed */
210 /* in the innermost loop which could lead to less cache misses. */
212 #define CN(I,J,K) cn[(I)*uinc+(J)*dim+(K)]
213 #define DCN(I, J) dcn[(I)*dcuinc+(J)]
215 if (uorder
== vorder
) {
216 for (k
= 0; k
< dim
; k
++) {
217 /* Derivative direction in u */
218 du
[k
] = vs
* (CN(1, 0, k
) - CN(0, 0, k
)) +
219 v
* (CN(1, 1, k
) - CN(0, 1, k
));
221 /* Derivative direction in v */
222 dv
[k
] = us
* (CN(0, 1, k
) - CN(0, 0, k
)) +
223 u
* (CN(1, 1, k
) - CN(1, 0, k
));
225 /* bilinear de Casteljau step */
226 out
[k
] = us
* (vs
* CN(0, 0, k
) + v
* CN(0, 1, k
)) +
227 u
* (vs
* CN(1, 0, k
) + v
* CN(1, 1, k
));
230 else if (minorder
== uorder
) {
231 for (k
= 0; k
< dim
; k
++) {
232 /* bilinear de Casteljau step */
233 DCN(1, 0) = CN(1, 0, k
) - CN(0, 0, k
);
234 DCN(0, 0) = us
* CN(0, 0, k
) + u
* CN(1, 0, k
);
236 for (j
= 0; j
< vorder
- 1; j
++) {
237 /* for the derivative in u */
238 DCN(1, j
+ 1) = CN(1, j
+ 1, k
) - CN(0, j
+ 1, k
);
239 DCN(1, j
) = vs
* DCN(1, j
) + v
* DCN(1, j
+ 1);
241 /* for the `point' */
242 DCN(0, j
+ 1) = us
* CN(0, j
+ 1, k
) + u
* CN(1, j
+ 1, k
);
243 DCN(0, j
) = vs
* DCN(0, j
) + v
* DCN(0, j
+ 1);
246 /* remaining linear de Casteljau steps until the second last step */
247 for (h
= minorder
; h
< vorder
- 1; h
++)
248 for (j
= 0; j
< vorder
- h
; j
++) {
249 /* for the derivative in u */
250 DCN(1, j
) = vs
* DCN(1, j
) + v
* DCN(1, j
+ 1);
252 /* for the `point' */
253 DCN(0, j
) = vs
* DCN(0, j
) + v
* DCN(0, j
+ 1);
256 /* derivative direction in v */
257 dv
[k
] = DCN(0, 1) - DCN(0, 0);
259 /* derivative direction in u */
260 du
[k
] = vs
* DCN(1, 0) + v
* DCN(1, 1);
262 /* last linear de Casteljau step */
263 out
[k
] = vs
* DCN(0, 0) + v
* DCN(0, 1);
266 else { /* minorder == vorder */
268 for (k
= 0; k
< dim
; k
++) {
269 /* bilinear de Casteljau step */
270 DCN(0, 1) = CN(0, 1, k
) - CN(0, 0, k
);
271 DCN(0, 0) = vs
* CN(0, 0, k
) + v
* CN(0, 1, k
);
272 for (i
= 0; i
< uorder
- 1; i
++) {
273 /* for the derivative in v */
274 DCN(i
+ 1, 1) = CN(i
+ 1, 1, k
) - CN(i
+ 1, 0, k
);
275 DCN(i
, 1) = us
* DCN(i
, 1) + u
* DCN(i
+ 1, 1);
277 /* for the `point' */
278 DCN(i
+ 1, 0) = vs
* CN(i
+ 1, 0, k
) + v
* CN(i
+ 1, 1, k
);
279 DCN(i
, 0) = us
* DCN(i
, 0) + u
* DCN(i
+ 1, 0);
282 /* remaining linear de Casteljau steps until the second last step */
283 for (h
= minorder
; h
< uorder
- 1; h
++)
284 for (i
= 0; i
< uorder
- h
; i
++) {
285 /* for the derivative in v */
286 DCN(i
, 1) = us
* DCN(i
, 1) + u
* DCN(i
+ 1, 1);
288 /* for the `point' */
289 DCN(i
, 0) = us
* DCN(i
, 0) + u
* DCN(i
+ 1, 0);
292 /* derivative direction in u */
293 du
[k
] = DCN(1, 0) - DCN(0, 0);
295 /* derivative direction in v */
296 dv
[k
] = us
* DCN(0, 1) + u
* DCN(1, 1);
298 /* last linear de Casteljau step */
299 out
[k
] = us
* DCN(0, 0) + u
* DCN(1, 0);
303 else if (uorder
== vorder
) {
304 for (k
= 0; k
< dim
; k
++) {
305 /* first bilinear de Casteljau step */
306 for (i
= 0; i
< uorder
- 1; i
++) {
307 DCN(i
, 0) = us
* CN(i
, 0, k
) + u
* CN(i
+ 1, 0, k
);
308 for (j
= 0; j
< vorder
- 1; j
++) {
309 DCN(i
, j
+ 1) = us
* CN(i
, j
+ 1, k
) + u
* CN(i
+ 1, j
+ 1, k
);
310 DCN(i
, j
) = vs
* DCN(i
, j
) + v
* DCN(i
, j
+ 1);
314 /* remaining bilinear de Casteljau steps until the second last step */
315 for (h
= 2; h
< minorder
- 1; h
++)
316 for (i
= 0; i
< uorder
- h
; i
++) {
317 DCN(i
, 0) = us
* DCN(i
, 0) + u
* DCN(i
+ 1, 0);
318 for (j
= 0; j
< vorder
- h
; j
++) {
319 DCN(i
, j
+ 1) = us
* DCN(i
, j
+ 1) + u
* DCN(i
+ 1, j
+ 1);
320 DCN(i
, j
) = vs
* DCN(i
, j
) + v
* DCN(i
, j
+ 1);
324 /* derivative direction in u */
325 du
[k
] = vs
* (DCN(1, 0) - DCN(0, 0)) + v
* (DCN(1, 1) - DCN(0, 1));
327 /* derivative direction in v */
328 dv
[k
] = us
* (DCN(0, 1) - DCN(0, 0)) + u
* (DCN(1, 1) - DCN(1, 0));
330 /* last bilinear de Casteljau step */
331 out
[k
] = us
* (vs
* DCN(0, 0) + v
* DCN(0, 1)) +
332 u
* (vs
* DCN(1, 0) + v
* DCN(1, 1));
335 else if (minorder
== uorder
) {
336 for (k
= 0; k
< dim
; k
++) {
337 /* first bilinear de Casteljau step */
338 for (i
= 0; i
< uorder
- 1; i
++) {
339 DCN(i
, 0) = us
* CN(i
, 0, k
) + u
* CN(i
+ 1, 0, k
);
340 for (j
= 0; j
< vorder
- 1; j
++) {
341 DCN(i
, j
+ 1) = us
* CN(i
, j
+ 1, k
) + u
* CN(i
+ 1, j
+ 1, k
);
342 DCN(i
, j
) = vs
* DCN(i
, j
) + v
* DCN(i
, j
+ 1);
346 /* remaining bilinear de Casteljau steps until the second last step */
347 for (h
= 2; h
< minorder
- 1; h
++)
348 for (i
= 0; i
< uorder
- h
; i
++) {
349 DCN(i
, 0) = us
* DCN(i
, 0) + u
* DCN(i
+ 1, 0);
350 for (j
= 0; j
< vorder
- h
; j
++) {
351 DCN(i
, j
+ 1) = us
* DCN(i
, j
+ 1) + u
* DCN(i
+ 1, j
+ 1);
352 DCN(i
, j
) = vs
* DCN(i
, j
) + v
* DCN(i
, j
+ 1);
356 /* last bilinear de Casteljau step */
357 DCN(2, 0) = DCN(1, 0) - DCN(0, 0);
358 DCN(0, 0) = us
* DCN(0, 0) + u
* DCN(1, 0);
359 for (j
= 0; j
< vorder
- 1; j
++) {
360 /* for the derivative in u */
361 DCN(2, j
+ 1) = DCN(1, j
+ 1) - DCN(0, j
+ 1);
362 DCN(2, j
) = vs
* DCN(2, j
) + v
* DCN(2, j
+ 1);
364 /* for the `point' */
365 DCN(0, j
+ 1) = us
* DCN(0, j
+ 1) + u
* DCN(1, j
+ 1);
366 DCN(0, j
) = vs
* DCN(0, j
) + v
* DCN(0, j
+ 1);
369 /* remaining linear de Casteljau steps until the second last step */
370 for (h
= minorder
; h
< vorder
- 1; h
++)
371 for (j
= 0; j
< vorder
- h
; j
++) {
372 /* for the derivative in u */
373 DCN(2, j
) = vs
* DCN(2, j
) + v
* DCN(2, j
+ 1);
375 /* for the `point' */
376 DCN(0, j
) = vs
* DCN(0, j
) + v
* DCN(0, j
+ 1);
379 /* derivative direction in v */
380 dv
[k
] = DCN(0, 1) - DCN(0, 0);
382 /* derivative direction in u */
383 du
[k
] = vs
* DCN(2, 0) + v
* DCN(2, 1);
385 /* last linear de Casteljau step */
386 out
[k
] = vs
* DCN(0, 0) + v
* DCN(0, 1);
389 else { /* minorder == vorder */
391 for (k
= 0; k
< dim
; k
++) {
392 /* first bilinear de Casteljau step */
393 for (i
= 0; i
< uorder
- 1; i
++) {
394 DCN(i
, 0) = us
* CN(i
, 0, k
) + u
* CN(i
+ 1, 0, k
);
395 for (j
= 0; j
< vorder
- 1; j
++) {
396 DCN(i
, j
+ 1) = us
* CN(i
, j
+ 1, k
) + u
* CN(i
+ 1, j
+ 1, k
);
397 DCN(i
, j
) = vs
* DCN(i
, j
) + v
* DCN(i
, j
+ 1);
401 /* remaining bilinear de Casteljau steps until the second last step */
402 for (h
= 2; h
< minorder
- 1; h
++)
403 for (i
= 0; i
< uorder
- h
; i
++) {
404 DCN(i
, 0) = us
* DCN(i
, 0) + u
* DCN(i
+ 1, 0);
405 for (j
= 0; j
< vorder
- h
; j
++) {
406 DCN(i
, j
+ 1) = us
* DCN(i
, j
+ 1) + u
* DCN(i
+ 1, j
+ 1);
407 DCN(i
, j
) = vs
* DCN(i
, j
) + v
* DCN(i
, j
+ 1);
411 /* last bilinear de Casteljau step */
412 DCN(0, 2) = DCN(0, 1) - DCN(0, 0);
413 DCN(0, 0) = vs
* DCN(0, 0) + v
* DCN(0, 1);
414 for (i
= 0; i
< uorder
- 1; i
++) {
415 /* for the derivative in v */
416 DCN(i
+ 1, 2) = DCN(i
+ 1, 1) - DCN(i
+ 1, 0);
417 DCN(i
, 2) = us
* DCN(i
, 2) + u
* DCN(i
+ 1, 2);
419 /* for the `point' */
420 DCN(i
+ 1, 0) = vs
* DCN(i
+ 1, 0) + v
* DCN(i
+ 1, 1);
421 DCN(i
, 0) = us
* DCN(i
, 0) + u
* DCN(i
+ 1, 0);
424 /* remaining linear de Casteljau steps until the second last step */
425 for (h
= minorder
; h
< uorder
- 1; h
++)
426 for (i
= 0; i
< uorder
- h
; i
++) {
427 /* for the derivative in v */
428 DCN(i
, 2) = us
* DCN(i
, 2) + u
* DCN(i
+ 1, 2);
430 /* for the `point' */
431 DCN(i
, 0) = us
* DCN(i
, 0) + u
* DCN(i
+ 1, 0);
434 /* derivative direction in u */
435 du
[k
] = DCN(1, 0) - DCN(0, 0);
437 /* derivative direction in v */
438 dv
[k
] = us
* DCN(0, 2) + u
* DCN(1, 2);
440 /* last linear de Casteljau step */
441 out
[k
] = us
* DCN(0, 0) + u
* DCN(1, 0);
450 * Do one-time initialization for evaluators.
453 _math_init_eval(void)
457 /* KW: precompute 1/x for useful x.
459 for (i
= 1; i
< MAX_EVAL_ORDER
; i
++)
460 inv_tab
[i
] = 1.0F
/ i
;